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<FONT color="green">001</FONT>    /*<a name="line.1"></a>
<FONT color="green">002</FONT>     * Licensed to the Apache Software Foundation (ASF) under one or more<a name="line.2"></a>
<FONT color="green">003</FONT>     * contributor license agreements.  See the NOTICE file distributed with<a name="line.3"></a>
<FONT color="green">004</FONT>     * this work for additional information regarding copyright ownership.<a name="line.4"></a>
<FONT color="green">005</FONT>     * The ASF licenses this file to You under the Apache License, Version 2.0<a name="line.5"></a>
<FONT color="green">006</FONT>     * (the "License"); you may not use this file except in compliance with<a name="line.6"></a>
<FONT color="green">007</FONT>     * the License.  You may obtain a copy of the License at<a name="line.7"></a>
<FONT color="green">008</FONT>     *<a name="line.8"></a>
<FONT color="green">009</FONT>     *      http://www.apache.org/licenses/LICENSE-2.0<a name="line.9"></a>
<FONT color="green">010</FONT>     *<a name="line.10"></a>
<FONT color="green">011</FONT>     * Unless required by applicable law or agreed to in writing, software<a name="line.11"></a>
<FONT color="green">012</FONT>     * distributed under the License is distributed on an "AS IS" BASIS,<a name="line.12"></a>
<FONT color="green">013</FONT>     * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.<a name="line.13"></a>
<FONT color="green">014</FONT>     * See the License for the specific language governing permissions and<a name="line.14"></a>
<FONT color="green">015</FONT>     * limitations under the License.<a name="line.15"></a>
<FONT color="green">016</FONT>     */<a name="line.16"></a>
<FONT color="green">017</FONT>    package org.apache.commons.math3.special;<a name="line.17"></a>
<FONT color="green">018</FONT>    <a name="line.18"></a>
<FONT color="green">019</FONT>    import org.apache.commons.math3.util.FastMath;<a name="line.19"></a>
<FONT color="green">020</FONT>    <a name="line.20"></a>
<FONT color="green">021</FONT>    /**<a name="line.21"></a>
<FONT color="green">022</FONT>     * This is a utility class that provides computation methods related to the<a name="line.22"></a>
<FONT color="green">023</FONT>     * error functions.<a name="line.23"></a>
<FONT color="green">024</FONT>     *<a name="line.24"></a>
<FONT color="green">025</FONT>     * @version $Id: Erf.java 1416643 2012-12-03 19:37:14Z tn $<a name="line.25"></a>
<FONT color="green">026</FONT>     */<a name="line.26"></a>
<FONT color="green">027</FONT>    public class Erf {<a name="line.27"></a>
<FONT color="green">028</FONT>    <a name="line.28"></a>
<FONT color="green">029</FONT>        /**<a name="line.29"></a>
<FONT color="green">030</FONT>         * The number {@code X_CRIT} is used by {@link #erf(double, double)} internally.<a name="line.30"></a>
<FONT color="green">031</FONT>         * This number solves {@code erf(x)=0.5} within 1ulp.<a name="line.31"></a>
<FONT color="green">032</FONT>         * More precisely, the current implementations of<a name="line.32"></a>
<FONT color="green">033</FONT>         * {@link #erf(double)} and {@link #erfc(double)} satisfy:&lt;br/&gt;<a name="line.33"></a>
<FONT color="green">034</FONT>         * {@code erf(X_CRIT) &lt; 0.5},&lt;br/&gt;<a name="line.34"></a>
<FONT color="green">035</FONT>         * {@code erf(Math.nextUp(X_CRIT) &gt; 0.5},&lt;br/&gt;<a name="line.35"></a>
<FONT color="green">036</FONT>         * {@code erfc(X_CRIT) = 0.5}, and&lt;br/&gt;<a name="line.36"></a>
<FONT color="green">037</FONT>         * {@code erfc(Math.nextUp(X_CRIT) &lt; 0.5}<a name="line.37"></a>
<FONT color="green">038</FONT>         */<a name="line.38"></a>
<FONT color="green">039</FONT>        private static final double X_CRIT = 0.4769362762044697;<a name="line.39"></a>
<FONT color="green">040</FONT>    <a name="line.40"></a>
<FONT color="green">041</FONT>        /**<a name="line.41"></a>
<FONT color="green">042</FONT>         * Default constructor.  Prohibit instantiation.<a name="line.42"></a>
<FONT color="green">043</FONT>         */<a name="line.43"></a>
<FONT color="green">044</FONT>        private Erf() {}<a name="line.44"></a>
<FONT color="green">045</FONT>    <a name="line.45"></a>
<FONT color="green">046</FONT>        /**<a name="line.46"></a>
<FONT color="green">047</FONT>         * Returns the error function.<a name="line.47"></a>
<FONT color="green">048</FONT>         *<a name="line.48"></a>
<FONT color="green">049</FONT>         * &lt;p&gt;erf(x) = 2/&amp;radic;&amp;pi; &lt;sub&gt;0&lt;/sub&gt;&amp;int;&lt;sup&gt;x&lt;/sup&gt; e&lt;sup&gt;-t&lt;sup&gt;2&lt;/sup&gt;&lt;/sup&gt;dt &lt;/p&gt;<a name="line.49"></a>
<FONT color="green">050</FONT>         *<a name="line.50"></a>
<FONT color="green">051</FONT>         * &lt;p&gt;This implementation computes erf(x) using the<a name="line.51"></a>
<FONT color="green">052</FONT>         * {@link Gamma#regularizedGammaP(double, double, double, int) regularized gamma function},<a name="line.52"></a>
<FONT color="green">053</FONT>         * following &lt;a href="http://mathworld.wolfram.com/Erf.html"&gt; Erf&lt;/a&gt;, equation (3)&lt;/p&gt;<a name="line.53"></a>
<FONT color="green">054</FONT>         *<a name="line.54"></a>
<FONT color="green">055</FONT>         * &lt;p&gt;The value returned is always between -1 and 1 (inclusive).<a name="line.55"></a>
<FONT color="green">056</FONT>         * If {@code abs(x) &gt; 40}, then {@code erf(x)} is indistinguishable from<a name="line.56"></a>
<FONT color="green">057</FONT>         * either 1 or -1 as a double, so the appropriate extreme value is returned.<a name="line.57"></a>
<FONT color="green">058</FONT>         * &lt;/p&gt;<a name="line.58"></a>
<FONT color="green">059</FONT>         *<a name="line.59"></a>
<FONT color="green">060</FONT>         * @param x the value.<a name="line.60"></a>
<FONT color="green">061</FONT>         * @return the error function erf(x)<a name="line.61"></a>
<FONT color="green">062</FONT>         * @throws org.apache.commons.math3.exception.MaxCountExceededException<a name="line.62"></a>
<FONT color="green">063</FONT>         * if the algorithm fails to converge.<a name="line.63"></a>
<FONT color="green">064</FONT>         * @see Gamma#regularizedGammaP(double, double, double, int)<a name="line.64"></a>
<FONT color="green">065</FONT>         */<a name="line.65"></a>
<FONT color="green">066</FONT>        public static double erf(double x) {<a name="line.66"></a>
<FONT color="green">067</FONT>            if (FastMath.abs(x) &gt; 40) {<a name="line.67"></a>
<FONT color="green">068</FONT>                return x &gt; 0 ? 1 : -1;<a name="line.68"></a>
<FONT color="green">069</FONT>            }<a name="line.69"></a>
<FONT color="green">070</FONT>            final double ret = Gamma.regularizedGammaP(0.5, x * x, 1.0e-15, 10000);<a name="line.70"></a>
<FONT color="green">071</FONT>            return x &lt; 0 ? -ret : ret;<a name="line.71"></a>
<FONT color="green">072</FONT>        }<a name="line.72"></a>
<FONT color="green">073</FONT>    <a name="line.73"></a>
<FONT color="green">074</FONT>        /**<a name="line.74"></a>
<FONT color="green">075</FONT>         * Returns the complementary error function.<a name="line.75"></a>
<FONT color="green">076</FONT>         *<a name="line.76"></a>
<FONT color="green">077</FONT>         * &lt;p&gt;erfc(x) = 2/&amp;radic;&amp;pi; &lt;sub&gt;x&lt;/sub&gt;&amp;int;&lt;sup&gt;&amp;infin;&lt;/sup&gt; e&lt;sup&gt;-t&lt;sup&gt;2&lt;/sup&gt;&lt;/sup&gt;dt<a name="line.77"></a>
<FONT color="green">078</FONT>         * &lt;br/&gt;<a name="line.78"></a>
<FONT color="green">079</FONT>         *    = 1 - {@link #erf(double) erf(x)} &lt;/p&gt;<a name="line.79"></a>
<FONT color="green">080</FONT>         *<a name="line.80"></a>
<FONT color="green">081</FONT>         * &lt;p&gt;This implementation computes erfc(x) using the<a name="line.81"></a>
<FONT color="green">082</FONT>         * {@link Gamma#regularizedGammaQ(double, double, double, int) regularized gamma function},<a name="line.82"></a>
<FONT color="green">083</FONT>         * following &lt;a href="http://mathworld.wolfram.com/Erf.html"&gt; Erf&lt;/a&gt;, equation (3).&lt;/p&gt;<a name="line.83"></a>
<FONT color="green">084</FONT>         *<a name="line.84"></a>
<FONT color="green">085</FONT>         * &lt;p&gt;The value returned is always between 0 and 2 (inclusive).<a name="line.85"></a>
<FONT color="green">086</FONT>         * If {@code abs(x) &gt; 40}, then {@code erf(x)} is indistinguishable from<a name="line.86"></a>
<FONT color="green">087</FONT>         * either 0 or 2 as a double, so the appropriate extreme value is returned.<a name="line.87"></a>
<FONT color="green">088</FONT>         * &lt;/p&gt;<a name="line.88"></a>
<FONT color="green">089</FONT>         *<a name="line.89"></a>
<FONT color="green">090</FONT>         * @param x the value<a name="line.90"></a>
<FONT color="green">091</FONT>         * @return the complementary error function erfc(x)<a name="line.91"></a>
<FONT color="green">092</FONT>         * @throws org.apache.commons.math3.exception.MaxCountExceededException<a name="line.92"></a>
<FONT color="green">093</FONT>         * if the algorithm fails to converge.<a name="line.93"></a>
<FONT color="green">094</FONT>         * @see Gamma#regularizedGammaQ(double, double, double, int)<a name="line.94"></a>
<FONT color="green">095</FONT>         * @since 2.2<a name="line.95"></a>
<FONT color="green">096</FONT>         */<a name="line.96"></a>
<FONT color="green">097</FONT>        public static double erfc(double x) {<a name="line.97"></a>
<FONT color="green">098</FONT>            if (FastMath.abs(x) &gt; 40) {<a name="line.98"></a>
<FONT color="green">099</FONT>                return x &gt; 0 ? 0 : 2;<a name="line.99"></a>
<FONT color="green">100</FONT>            }<a name="line.100"></a>
<FONT color="green">101</FONT>            final double ret = Gamma.regularizedGammaQ(0.5, x * x, 1.0e-15, 10000);<a name="line.101"></a>
<FONT color="green">102</FONT>            return x &lt; 0 ? 2 - ret : ret;<a name="line.102"></a>
<FONT color="green">103</FONT>        }<a name="line.103"></a>
<FONT color="green">104</FONT>    <a name="line.104"></a>
<FONT color="green">105</FONT>        /**<a name="line.105"></a>
<FONT color="green">106</FONT>         * Returns the difference between erf(x1) and erf(x2).<a name="line.106"></a>
<FONT color="green">107</FONT>         *<a name="line.107"></a>
<FONT color="green">108</FONT>         * The implementation uses either erf(double) or erfc(double)<a name="line.108"></a>
<FONT color="green">109</FONT>         * depending on which provides the most precise result.<a name="line.109"></a>
<FONT color="green">110</FONT>         *<a name="line.110"></a>
<FONT color="green">111</FONT>         * @param x1 the first value<a name="line.111"></a>
<FONT color="green">112</FONT>         * @param x2 the second value<a name="line.112"></a>
<FONT color="green">113</FONT>         * @return erf(x2) - erf(x1)<a name="line.113"></a>
<FONT color="green">114</FONT>         */<a name="line.114"></a>
<FONT color="green">115</FONT>        public static double erf(double x1, double x2) {<a name="line.115"></a>
<FONT color="green">116</FONT>            if(x1 &gt; x2) {<a name="line.116"></a>
<FONT color="green">117</FONT>                return -erf(x2, x1);<a name="line.117"></a>
<FONT color="green">118</FONT>            }<a name="line.118"></a>
<FONT color="green">119</FONT>    <a name="line.119"></a>
<FONT color="green">120</FONT>            return<a name="line.120"></a>
<FONT color="green">121</FONT>            x1 &lt; -X_CRIT ?<a name="line.121"></a>
<FONT color="green">122</FONT>                x2 &lt; 0.0 ?<a name="line.122"></a>
<FONT color="green">123</FONT>                    erfc(-x2) - erfc(-x1) :<a name="line.123"></a>
<FONT color="green">124</FONT>                    erf(x2) - erf(x1) :<a name="line.124"></a>
<FONT color="green">125</FONT>                x2 &gt; X_CRIT &amp;&amp; x1 &gt; 0.0 ?<a name="line.125"></a>
<FONT color="green">126</FONT>                    erfc(x1) - erfc(x2) :<a name="line.126"></a>
<FONT color="green">127</FONT>                    erf(x2) - erf(x1);<a name="line.127"></a>
<FONT color="green">128</FONT>        }<a name="line.128"></a>
<FONT color="green">129</FONT>    }<a name="line.129"></a>
<FONT color="green">130</FONT>    <a name="line.130"></a>




























































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